On the use of fractional calculus to improve the pulse arrival time (PAT) detection when using photoplethysmography (PPG) and electrocardiography (ECG) signals

The pulse arrival time (PAT) has been considered a surrogate measure for pulse wave velocity (PWV), although some studies have noted that this parameter is not accurate enough. Moreover, the inter-beat interval (IBI) time series obtained from successive pulse wave arrivals can be employed as a surrogate measure of the RR time series avoiding the use of electrocardiogram (ECG) signals. Pulse arrival detection is a procedure needed for both PAT and IBI measurements and depends on the proper fiducial points chosen. In this paper, a new set of fiducial points that can be tailored using several optimization criteria is proposed to improve the detection of successive pulse arrivals. This set is based on the location of local maxima and minima in the systolic rise of the pulse wave after fractional differintegration of the signal. Several optimization criteria have been proposed and applied to high-quality recordings of a database with subjects who were breathing at different rates while sitting or standing. When a proper fractional differintegration order is selected by using the RR time series as a reference, the agreement between the obtained IBI and RR is better than that for other state-of-the-art fiducial points. This work tested seven different traditional fiducial points. For the agreement analysis, the median standard deviation of the difference between the IBI and RR time series is 5.72 ms for the proposed fiducial point versus 6.20 ms for the best-performing traditional fiducial point, although it can reach as high as 9.93 ms for another traditional fiducial point. Other optimization criteria aim to reduce the standard deviation of the PAT (7.21 ms using the proposed fiducial point versus 8.22 ms to 15.4 ms for the best- and worst-performing traditional fiducial points) or to minimize the standard deviation of the PAT attributable to breathing (3.44 ms using the proposed fiducial point versus 4.40 ms to 5.12 ms for best- and worst-performing traditional fiducial points). The use of these fiducial points may help to better quantify the beat-to-beat PAT variability and IBI time series.


Introduction
The pulse arrival time (PAT) is becoming a useful measure of the cardiovascular system [1,2] due to its convenient and noninvasive nature.PAT, for one cardiac cycle, is defined as the interval between the apex of the R wave measured using an electrocardiogram lead and the arrival of the pulse wave (pulse arrival (PA)) at a distal site, which is generally measured by plethysmographic methods (for example, using a photoplethysmographic pulse sensor attached to the wrist, finger, forehead, or ear).The PAT is used as a surrogate measure of the pulse wave transit time (PTT) which is defined as the time that the pulse wave elapses to travel between two locations, one downstream from the other [1].If the location proximal to the heart is the aortic valve and the distal location is the same as that used to evaluate the PAT, both intervals can be related by: The pre-ejection period (PEP(n)) is clinically defined from the start of the Q wave to the opening of the aortic valve [3] of the n th measured cardiac cycle.Electromechanical delay (EMD) is defined from the start of ventricular depolarization (Q wave) to the beginning of the increase in pressure in the left ventricle and healthy subjects are in the range between 30 ms and 40 ms [4].The EMD can be considered constant for the same individual so we assume that it does not change on a beat-to-beat basis [5].Moreover, the QR interval (derived from the QRS complex) can also be considered a constant for the same individual during normal measuring intervals [6].On the other hand, the isovolumetric contraction (IVC) interval corresponds to the increase in pressure in the left ventricle and ends with the opening of the aortic valve.This interval is regulated by the autonomic nervous system (ANS) [7]; hence, it has beatto-beat variability.In summary, the beat-to-beat difference between PAT and PTT can be attributed to beat-to-beat changes in the PEP that are attributable to changes in the IVC.Kortekaas MC et al. [8] measured the contribution of the variability in the PEP to the variability of PAT and concluded that it was approximately 20% of the total variability.
The PEP increases when changing posture from supine to standing [5,6,9], decreases with physical activity [6,10], and does not depend on its average value concerning respiratory frequency [9].The PEP is highly modulated by mental and physical load [10].The PEP is regarded as an indicator of sympathetic beta-adrenergic control of the heart period but only in stationary conditions, such as when posture does not change during the measurement [9].If posture remains constant, an increase in the PEP can be interpreted as an increase in sympathetic beta-adrenergic activity.Because ventricular contractility and, accordingly, PEP is regulated by the sympathetic arch, it is reasonable to think that beat-to-beat changes in the PEP can be modeled as a low-frequency (LF) process with relevant spectral components under 0.1 Hz.Miyano H et al. [11] showed that in isolated dog hearts, the transfer function of changes in ventricular contractility induced by sympathetic stimulation at frequencies up to 5 Hz behaves as a lowpass system with decreasing magnitudes from frequencies as small as 0.02 Hz.Hence, it is expected that beat-to-beat changes in the PEP (when no other sources of uncertainty in its measurement are present) will smooth.
On the other hand, PTT is commonly used as a noninvasive method for estimating blood pressure and arterial stiffness.Several studies, such as [12 and 13], have stressed that PTT (or pulse wave velocity (PWV)) is more strongly correlated with systolic, mean, and diastolic blood pressure than is PAT.Hence, predictions of blood pressure from PTT measurements are more accurate than from PAT measurements.The cause of this lack of correlation using PAT is the aforementioned variability in PEP.Moreover, some studies have indicated that PTT and the PEP are not independent [13].Despite all these complex relationships among cardiac intervals, some studies have recognized that the PAT could be employed as a reasonable surrogate for blood pressure estimation [14].The lack of consensus among studies can be attributed not only to the physiological variabilities of PEP and PTT but also to inaccuracies in the measurement of the different cardiac intervals.
Although the R wave of an electrocardiogram (ECG) is a sharp and well-defined fiducial point, the pulse arrival at a distal site is not so well-defined.Hence, we can model the PAT of the n th beat in a recording as follows: where PA ideal (n) is the detected pulse arrival at the distal site using an ideal fiducial point detection algorithm that can change on a beat-to-beat basis due to physiological modulations of the pulse wave velocity and PEP.In (2), ν PA (n) is a stochastic process associated with the inaccuracy of real fiducial point detection and R(n) is the location of the R wave (we suppose that the time location uncertainty in the detection of the R wave is negligible before the detection of pulse arrival).
On the other hand, PTT and the PEP can be written as follows: PEPðnÞ ¼ Ao ideal ðnÞ þ v Ao ðnÞ À ðRðnÞ À QRÞ ð4Þ where Ao ideal (n) corresponds to an ideal fiducial point detection of the aortic valve opening and includes feasible physiological beat-to-beat modulations; ν Ao (n) is the inaccuracy modeling process associated with a real aortic valve opening detector, and QR is supposed to be constant.On the other hand, if the PTT is measured using the pulse wave at two sites (proximal and distal), which is the common way to assess PWV due to the difficulty of measuring aortic valve opening, then the PTT is defined as follows: Eqs (1) to (5) stress that all the different intervals change from beat to beat.While measuring these intervals, what we want to retain are changes in physiological origin and are not associated with beat-to-beat inaccuracies in fiducial point determination.PAT and/or PTT variability analysis [15,16,17] is still an emergent field for cardiovascular control research, probably because of the low variance of the obtained time series (when measured without inducing physiological changes).The low variance of the signal dictates that inaccuracies in fiducial point detection should be very low to ensure a fair signal-to-noise ratio.Identifying a sufficiently accurate fiducial point could provide better estimations of beat-to-beat PAT and PTT time series, hence improving the ability of these analyses to gain better noninvasive knowledge of cardiovascular regulation.
On the other hand, the inter-beat interval (IBI), defined as the time series obtained by measuring the interval between consecutive PAs, has been proposed as a surrogate measure of the RR time series obtained from the successive intervals between consecutive R waves measured using ECG [18,19].As identified in some studies, fiducial point selection for pulse arrival detection has an impact on the agreement between the IBI and RR.We can write: Hence, the differences between the two-time series that quantify their agreement depend on the accuracy of the detection of successive pulse arrivals.
Regardless of the application (PAT or PTT variability analysis, RR vs. IBI comparisons), fiducial point detection is a key factor that undermines the accuracy of measurements.Chiu YC et al. [20] compared several of the most commonly employed fiducial points for PA detection for pulse wave velocity (PWV) measurements: the foot of the pulse wave, the point with the maximum first derivative, the point with the maximum second derivative, and the point where the tangent lines through the foot and maximum derivative points intercept.This same study did not consider the peak of the pulse wave due to its sensitivity to the presence of wave reflections although the peak has been considered the principal or auxiliary fiducial point in many other studies [18], [21,22,23].Previously we concluded [24] that for PWV determination, the best fiducial points are the tangent method [25] and the point at the maximum of the second derivative.Several other studies, such as [25], have recognized the impact of noise on fiducial point stability.Solà J et al. [26] proposed a parametric estimation of the pulse wave to obtain a fiducial point for each pulse wave arrival.This study revealed that the best fiducial points that provide a PAT with the greatest repeatability (i.e., providing a PAT time series with the lowest variability) are the fiducial points obtained from fitting the rising edge of the pulse to a hyperbolic tangent model (TANH), the maximum of the second derivative and the tangent method.In a previous work [24], we found that to maximize the agreement between the RR and the IBI time series obtained from a smartphone camera, the best fiducial point, among the foot, the maximum of the second derivative, and the maximum of the first derivative, was the last fiducial point when proper filtering of the photoplethysmogram (PPG) was performed.Hence, the choice of fiducial point detection depends on the intended application of the PAT measurement.
The set of fiducial points available in the previously mentioned (and other) studies is limited: a fiducial point is obtained by looking for a local maximum (or minimum) in the PPG after some processing or filtering has been performed.The resulting post-processed signal can be the difference between two fitted straight lines (tangent method), the peak or trough of a bandpass filtered version of the acquired PPG, the maximum of the first or second derivative of the filtered PPG, or the minimum of an error function between the filtered PPG and a fitting model (TANH method).Accordingly, previous research and the current study recognize the importance of accurate fiducial point detection for obtaining reliable PAT and PTT measurements.This study attempts to present a comprehensive set of fiducial points for PAT detection, optimize the accuracy of these selected fiducial points, and systematically compare these points with previously established points.Specifically, this study addresses specific challenges, such as the unpredictability of PEP variations (which leads to an increase in complexity when considering the interaction between PEP and IVC [8]) and the unclear nature of pulse arrival, and emphasizes practical applications, such as PAT variability analysis and RR vs. IBI comparisons.Furthermore, a comparison between PAT and PWV indicates that PWV has a stronger correlation with blood pressure, suggesting that PAT has limitations related to PEP fluctuation and the complex interaction between PEP and PTT [27].Identifying accurate fiducial points is another challenge since the arrival of the pulse is not as clear as the distinct R wave seen in electrocardiograms.Consequently, the proposed model emphasizes the necessity of distinguishing ideal and actual detection methods, acknowledging the uncertain nature of real fiducial point detection.The need for precise detection to improve the accuracy of PAT and PTT measures in cardiovascular research and monitoring is highlighted by the crucial role that fiducial points play in measurements, as has been studied in several previous studies [24][25][26].As a result, this study aimed to design a new approach for PAT detection, to address these challenges and improve the reliability of cardiovascular assessments.This study emphasizes the effectiveness of our proposed set of fiducial points in optimizing key objective criteria.Specifically, our findings show the potential to minimize agreement between the RR and IBI time series, as well as minimize variance in the PAT time series.Nevertheless, further investigations, possibly involving more extensive and potentially invasive studies, are imperative to determine the most suitable criterion for maximizing agreement between PAT and crucial cardiovascular parameters such as blood pressure or arterial stiffness.

Materials and methods
In this section, we introduce a comprehensive set of fiducial points derived from identifying local maxima and minima following the fractional differintegration of the pulse wave.These points are then applied to high-quality recordings from a database, monitoring subjects exhibiting various breathing patterns.Subsequently, we determine the optimal fractional order for each recording using diverse optimization criteria.We then compare the performances of these optimal orders with those of other reference points related to pulse arrival detection that have been documented in the literature.

Fiducial point based on fractional differintegration set proposal
Because the peak and foot of the pulse, the maximum of the first differentiation and the maximum of the second differentiation are commonly used fiducial points, a natural extension is to use the locations of local maxima and minima after computing the fractional differintegration of the systolic phase of the pulse wave for a large enough number of fractional orders as candidate fiducial points.The proposed fiducial points for pulse arrival time (PAT) measurements, derived from local maxima and minima in the systolic rise after fractional difference integration, offer several advantages over traditional methods.These methods demonstrate dynamic adaptability to varying pulse wave morphologies, increased specificity in localizing key events, and an improved signal-to-noise ratio.These points contribute to boosted beat-to-beat stability, reducing dependency on resting conditions and displaying potential for real-time applications.These fiducial points were chosen because of their physiological relevance and robust detection capabilities, as they can capture significant hemodynamic events during the systolic phase.This approach can reduce noise effects and maintains consistency between beats by considering local features, which are essential for reliable beat-to-beat analyses.
Fractional differintegration is an extension of differential calculus that has found practical applications in several fields of science [28].In [29], we used a fast algorithm to approximate the discrete fractional differintegration of a discrete signal that relies on the Gru ¨nwald-Letnikov approach, which approximates the fractional differintegration as described in [28] by: where x(n) is the n th sample of the signal to be fractional differintegrated, f s is the sampling frequency of the signal that is a scaling factor irrelevant for our analysis because we are interested in locating maxima and minima, Γ(k) is the gamma function evaluated at k 2R and α 2R is the fractional order.
For α = 1 the differentiation of the signal is obtained because For α = 2 the second differentiation of the signal is obtained as: For α = 0, the signal remains unchanged: For α = -1, the cumulative sum divided by the sampling frequency (integration) of the signal is obtained: This last expression uses Euler's reflection formula, which states: Hence, for j = n+1 and z = n+1 = j: For α = 2 Z, signals different from that those obtained from simple differentiation or cumulative sums are obtained.Decreasing the value of α provides smoother versions of the signal, while increasing the value of α enhances rapid changes in the signal.
Eq (7) has accuracy problems due to the exponential-like behavior of the gamma function.In this work, we use the approximation proposed in [30] to overcome that problem.The fractional differintegration can be approximated by a filter: where the coefficients of the filter are recursively obtained as follows: Fig 1 shows an example of a cycle of a pulse wave, its systolic rise, and some of its fractional differintegrated resulting signals.Note that by changing the value of α, the resulting signal may reach a maximum and/or a minimum during the systolic increase at different locations.We propose as an infinite set of candidate fiducial points for pulse arrival (PA) any maximum and minimum located inside the interval between the foot and peak of the pulse wave using a fractional differintegration of order α.In the following section, we will write PA a tag (n) to denote the fiducial point obtained for the beat n using the fractional differintegrated of the order α signal and locate, when the tag is 1, its maximum inside the systolic rise or its minimum when the tag is -1.For example, PA 0:57 À 1 (38) is the minimum of the fractional differintegrated signal using a fractional order of 0.57 that was found during the systolic increase in the 38 th beat of the recording.Some optimization criteria based on the dynamics of PA a tag (n) for all the available beats in the recording will allow us to optimize the values of α and the tag.

Experiment description
This work aimed to compare the performance of several fiducial points for PA estimation by comparing time series that employ these fiducial points using recordings made under stationary and still conditions.We will compare our proposed set of fiducial points obtained from fractional differintegration of the pulse wave with other fiducial points employed in previous literature outlined in the Introduction section.We chose a subset of a series of previously recorded physiological signals that were the basis of the experiment described in [21] and added some recordings that were not used for that paper.The complete set of recordings is available at [31] in the Open Science Framework (OSF) repository.
Participants.The dataset comes from 20 adult volunteers without known cardiac or respiratory diseases (15 male, 5 female) aged 22 to 56 years (mean age: 32.3 years, standard deviation: 11.8 years).Their weight and height expressed as the mean ± standard deviation, were 73.9 kg ± 13.0 kg and 173 cm ± 9 cm, respectively.Written informed consent was obtained from all participants involved in the study.The study was conducted according to the Data collection.For each subject, a session with a total of 9 recordings simultaneously sampling two ECG leads, PPG signals, and respiratory effort signals was carried out.Each recording lasted approximately 5 minutes.Six recordings corresponded to measurements while the subject was sitting, while the remaining three were made while the subject was standing.The position at the start of the session was chosen at random.The signals were collected at different breathing rates to assess the influence of breathing rate on the accuracy of the measurements.Respiratory sinus arrhythmia is related to the breathing rate [30].
In this study, we aimed to analyze, among other factors, the influence of this arrhythmia on the accuracy of heart rate variability (HRV) derived from the IBI and how different breathing rates may affect the PA.The breathing rates were 6 breaths per minute (bpm), 15 bpm, and free breathing (breathing at will).The 6 and 15 bpm breath frequencies were controlled using a visual aid presented to the subject via a smartphone application.While sitting, the subject breathed at 6 bpm during two recordings, at 15 bpm during another two recordings, and at will (free breathing) for the other two recordings.The order of the breathing rates was randomized.While standing, the subjects breathed in random order at 6 bpm, 15 bpm, and at will in each of the three recordings.After finishing the recordings for one posture, the subject changed actively to the other posture, and five minutes were allowed before the start of the remaining recordings.The details of the subjects, posture, and breathing rate for each recording are disclosed in [31].For two subjects, one of the nine measurements was considered unsuitable for measurement due to the intermittent disappearance of the pulse wave, both of which corresponded to the standing position.Accordingly, a total of 178 recordings with a length of 5 min are available in the repository.
Each recording in the dataset contains four physiological signals: two leads of the ECG, the PPG, and the respiratory signal.The ECG was measured using 6 skin electrodes (3 M Red Dot 2560) placed in standard lead I and standard lead II and two ECG (Biopac SS2LB) cables.The pulse wave was measured with a PPG sensor (Biopac SS4LA) placed in the middle finger of the right hand.Moreover, the respiratory signal was measured with a respiratory effort transducer (Biopac SS5LB).All four channels were connected to a Biopac MP36E acquisition system, where the signals were properly conditioned and acquired using a sampling rate of 5 kHz.For the ECG, the bandwidth of the Biopac amplifier was set between 0.05 Hz and 150 Hz.A 2 nd order Notch filter at 50 Hz was also included to remove main interference.For the PPG acquisition, a hardware high pass filter with a cutoff frequency of 0.05 Hz and a software low pass filter with a cutoff frequency of 30 Hz were used.The respiratory signal was acquired using the preset configuration recommended for the SS5LB sensor.

Data processing
In the data processing section, we detail the PAT and IBI measurements based on some optimization criteria.As a first step in the processing of the recordings, the standard lead I of the ECG was employed to detect the QRS waves using a custom QRS detector, which will be subsequently explained.
Beat detection in the ECG and PPG signals.The rough pulse arrivals (PA) in the PPG signal were also detected using a similar detector after proper filtering of the PPG.No ECG information was available for the pulse arrival detector, and no PPG information was available for the QRS detector.
The QRS detector is intended to be used in ECG signals with low disturbances such as noise, baseline drift, interference, or movement artifact while allowing the R wave amplitude to change due to breathing or other causes of QRS amplitude modulation.The ECG signals were first filtered using a bidirectional 4 th order Butterworth bandpass filter with cutoff frequencies between 10 Hz and 30 Hz.After filtering, a maximum filter was applied to the signal.The output of this filter for sample n is the maximum of the filtered ECG signal in the interval starting at n-W and finishing at n+W, where W is an input empirical parameter of the QRS detector.Most of the recordings provide accurate detection using W = 600 ms.The R peak locations correspond to the samples where the maximum filter output equals the value of the filtered ECG.
The pulse arrival detector for the PPG is quite similar.First, a bidirectional 4 th order Butterworth bandpass filter with empirically determined cutoff frequencies (between 2 Hz and 15 Hz enhances the systolic rise of the PPG for most morphologies) is applied.Next, a maximum filter with the same characteristics as the QRS detectors is used to locate the peaks of the filtered PPG.Fig 2 shows an example of both detectors.
All the MATLAB1 codes for both detectors, including the implementation of the maximum filter, is available at [31].R wave and rough pulse arrival detection were performed using these detectors while checking that the RR and IBI time series obtained from these detectors were similar to each other, from their physiological origin (by checking mean heart rate) and without (or with the smallest possible amount of) outliers.For a few ECG and PPG signals, the RR and IBI time series contained outliers when the detection was performed using the recommended W. For these recordings, the W parameter of the maximum filter was adjusted until the resulting time series was probed satisfactorily.For an even smaller set of IBI time series, the cutoff frequencies were also adjusted.At the end of this procedure, a set of R wave and pulse arrival (P) locations with very few outliers was obtained.These locations are also available for each recording in [31].Although the R wave location will be considered the fiducial point for the ECG against which the PA will be compared, the rough P locations will be the starting point for the definition of new PA fiducial points in the next sections.No outlier correction was performed at the R or P locations because accuracy measures automatically reject them, as will be later explained.Nevertheless, if the first detected P location occurs before the first detected R wave, this P location is rejected.Moreover, if the last R wave does not have a P location after it, this R wave location is also rejected.
Fiducial point set generation.Once rough PA fiducial points are obtained for each recording, this section presents the general approach to optimizing the PA fiducial points based on fractional differintegration using three different optimization criteria.These criteria aim to find an optimal α and tag for each recording to minimize the standard deviation of a certain time series.As we will see, each criterion will be different from another because of this time series definition.The differences between the optimized PA fiducial points according to the different criteria and among other PA fiducial point definitions (i.e., using the TANH or the foot of the pulse wave) will also be assessed.
For each recording, the first and last R and P locations were removed for analysis to avoid searching for fiducial points before the start or after the end of the recording.Before the fractional differintegration computation using Eq (14) and (15), the PPG was first bandpass filtered using a 4 th order bidirectional Butterworth filter with cutoff frequencies set at 0.5 Hz and 15 Hz.
Next, the peak of each pulse wave was obtained by determining the maximum of the filtered PPG signal in an interval starting at the P rough location (PA rough ) and ending 100 ms later.This peak will be one additional fiducial point for comparison of the results (PA peak ).Next, the foot of each pulse wave was obtained (PA foot ) by searching for the minimum of the filtered PPG in an interval starting 300 ms before the peak and ending at the peak location.The foot of each pulse will be another fiducial point for comparison with our proposed fiducial point.Once the foot and peak were located for each PA defining the start and end of the systolic rise, an interval starting 40 ms before the foot and ending at the peak was defined for each beat.This filtered PPG interval was fractional differintegrated using different α values, and the maximum or minimum location of the resulting signal inside the systolic rise was considered a new candidate fiducial point.We started the fractional differintegration 40 ms before the foot to mitigate the impulse response of the filter defined by Eqs ( 14) and (15).For each candidate fiducial point obtained using the same α and tag, a PA a tag (n) time series was obtained.For each recording, we tested 4002 different candidate fiducial point time series using an α from -10 (integration of tenth order) to 10 (differentiation of sixth order) in steps of 0.01 and the two values of tag (-1 for the search of local minima and 1 for local maxima).Hence, this procedure started with a PA rough (n) time series and produced a 4002 PA a tag (n) time series.Moreover, two additional time series for comparison (PA peak (n) and PA foot (n)) were obtained in intermediary steps.Nevertheless, if PA 0 1 (n) = PA peak (n) and PA 0 À 1 (n) = PA foot (n), then they are included in the set of time series considered for optimization.
For the sake of comparison, the peak, foot, and starting P rough locations, the next fiducial point time series was also computed: a.The maximum of the first differentiated filtered PPG signal (PA pfd (n) but also included in the candidate fiducial point time series set as PA 1 1 (n)).b.The maximum of the second differentiated filtered PPG signal (PA psd (n) was also included in the candidate fiducial point time series set as PA 2 1 (n)).c.The average location of the TANH fitting of the filtered PPG (PA tanh (n)).PPG with the foot level was also recorded (PA tan (n)).
The code for the computation of the different time series as well as the files containing the sets of the different fiducial points for each recording can be found in [31].
Optimization by maximizing the agreement of the IBI and RR time series.The first optimization consists of maximizing the agreement between the RR time series obtained from the successive intervals between R wave locations and the IBI time series obtained from the successive intervals between PA locations (maxagr optimization criterion).The beat-to-beat disagreement between both time series is defined as follows: Moreover, Hence, the disagreement between time series can also be interpreted as the differentiation of the PAT time series.
The maximization of agreement employed in this work consists of choosing a proper pair of α and tag for the fractional differintegrated derived fiducial points that minimize the standard deviation of the disagreement time series: For the estimation of the standard deviation (σ) of the disagreement time series, samples of the da(n) time series with absolute values higher than 300 ms were not included.To gain further insight into the dynamics of the time series after fiducial point optimization, the maximum agreement (or minimum disagreement) time series is defined as follows: and will be used for comparison to the corresponding time series obtained using other fiducial points.The PAT time series defined using PA a opt tag opt (n)-R(n) will also be considered for comparison with the PAT time series obtained using other fiducial points.
Optimization by minimization of the variance of the PAT time series.If we suppose that the v PA (n) process in Eq (2) is uncorrelated with the pulse arrival time, the variance in the pulse arrival time and the variance in the inaccuracy process provide good estimates of the total variance in the PAT time series.Hence, minimizing the variance of the PAT time series equals minimizing the variance of v PA (n) if PA(n) and v PA (n) are independent (while the inaccuracies in R detection can be neglected).The second optimization criterion consists of minimizing the standard deviation of the PAT time series (minSDPAT optimization criterion): For the estimation of the standard deviation (σ) of the PAT, those PAT samples with absolute values higher than 300 ms were not included.As with the previous criterion, the mPAT time series is defined as follows: mPATðnÞ ¼ PA a opt tag opt ðnÞ À RðnÞ ð21Þ will be used for comparison with those obtained using other fiducial points.
Optimization by minimizing the variance of the high pass filtered PAT time series.Under the hypothesis that PTT variability is regulated by changes in blood pressure and arterial stiffness and that both variables are regulated by the sympathetic nervous system, it is feasible to think that a PAT time series that mimics changes in PTT should have smooth dynamics.Hence, the minimization of the standard deviation of the PAT time series after high pass filtering using a 0.15 Hz cutoff frequency is proposed as a third optimization criterion (minSDHPF optimization criterion).
Because the PAT time series is unevenly sampled, to obtain the high pass filtered PAT time series (HPFPAT), the original PAT is resampled at 4 Hz using linear interpolation first obtaining an evenly sampled PAT time series (esPAT(k)).Those PAT intervals that were deemed outliers (those with a departure greater than 300 ms from the average PAT) were skipped from the interpolation procedure.Then, a bidirectional (linear phase) 4 th order high pass Butterworth filter, with a 0.15 Hz cutoff frequency is applied.Then, the applied optimization criterion is: where the operator HPF () denotes the interpolation and filtering procedure previously described.Like in the previous criteria, the smooth, evenly sampled PAT time series is defined as follows: and compared with the dynamics of the corresponding time series employing other fiducial points.The PAT time series defined using PA a opt tag opt ðnÞ À RðnÞ will also be considered for comparison with the PAT time series obtained using other fiducial points.
The code for the estimation of PA a opt tag opt for the different optimization criteria as well as the files containing the sets of the different fiducial points for each recording can be found in [31].
Statistical analysis of α opt and tag opt .After processing all the recordings and obtaining the different fiducial points, the first study among the PA fiducial points characterized the values of α opt and tag opt among the different optimization criteria.Because α opt is not normally distributed but has a long-tailed median, 10% and 90% percentiles were obtained.Pairwise paired sign tests were used to assess the statistical significance of the difference in the median α opt for the different optimization criteria.Because tag opt can only have two possible values, it follows a Bernoulli distribution.The percentage of times that tag opt is 1 (how many times the fiducial point corresponds to a maximum) is estimated for each optimization criterion.Once again, pairwise paired sign tests were used to assess the statistical significance of differences in tag opt according to the optimization criteria.Moreover, unpaired sign tests for α opt and tag opt for each optimization criterion were performed to assess whether the values ofα opt and tag opt showed statistically significant differences from zero median distributions.In the case of α opt , a statistically significant difference will show that the optimization criterion is more prone to obtaining fiducial points corresponding to fractional integrating or fractional differentiating the pulse wave; for tag opt , each optimization criterion will show if a maximum is more likely to be chosen instead of a minimum.In addition, nonparametric Fisher tests were employed to determine statistically significant nonrandom associations between the sign of α opt and tag opt for the different optimization criteria.Finally, regarding the relationships of the values of α opt and tag opt with several potential influential factors, the nonparametric Kruskal-Wallis test was applied for each optimization criterion to ascertain whether their values are affected by intersubject variability, breathing pattern, or posture.

Fiducial point comparison.
To compare among the fiducial points, several features of the PAT time series obtained using PA a opt tag opt for the three optimizations and those obtained using the other seven fiducial points were obtained.The main reason for selecting these seven fiducial points lies in their prevalence as commonly identified optimal points in the PPG waveform, as reported in previous studies.Hemon and Phillips [32] compared methods for deriving instantaneous pulse periods (PP) from PPG signals, finding the best correlation with RR intervals using the intersecting tangent method.Also, several studies used high-frequency features, such as the foot, for PAT estimation, emphasizing their superiority over low-frequency fiducial points [33,34].Moreover, PPG signal derivatives are employed to enhance the recognition of points of interest and facilitate waveform interpretation [35,36].However, applying these fiducial points directly to noisy and corrupted PPG signals often results in unreliable and erratic results, compromising the repeatability of clinical PAT measurements.Hence, sophisticated methods for PPG waveform feature detection, including multi-dimensional parametric estimation [26] and the diastole-patching method [37] have been developed.
Previously, for feature estimation, outliers in each PAT time series and for each fiducial point were detected and corrected.To do so, a median filter with a window of 11 samples was applied to each time series.Any departure of the time series from the median filtered time series higher than 50 ms was considered an outlier.The selected features were estimated by skipping the detected outliers.The percentage of outliers concerning the total number of PAT intervals was annotated for further reporting.
The selected features were the mean value of the PAT (PAT), the standard deviation of the PAT (σ PAT ), and an approximation of the contribution of respiratory activity to PAT variability (ResprPAT).To derive a time series representative of respiratory activity synchronous with the PAT and having the same number of samples, the respiratory signal was sampled at each R wave location to obtain the RsRESP time series as follows: where RESP(k) is the k th sample of the acquired respiratory signal as previously described and R(n) is the sample at which the n th beat out of N has been detected.For each analyzed PAT time series, the ResprPAT was estimated from the value of the cross-correlation function between the RsRESP and PAT time series evaluated at the delay that provides the maximum absolute value of cross-correlation as follows: The cross-correlation function was estimated as, The respiratory contribution to PAT was estimated as follows: where, and σ RsRESP is the sample standard deviation of the RsRESP time series.
To compare among fiducial points, for each of the three features (PAT,σ PAT and ResprPAT) and each fiducial point, the 10%, 50% (median) and 90% percentiles were first obtained.While a comparison of PAT among fiducial points provides information on the central location of the fiducial points over the systolic rise of the pulse wave (being the foot and the peak fiducial points in their most extreme cases), a comparison of σ PAT will reveal the contribution of each fiducial point to the total variability of the PAT.Finally, a comparison of the ResprPAT will reveal whether there are relevant differences in the contribution of respiratory parameters to PAT variability.To perform this comparison, pairwise paired sign tests were performed to assess significant differences in features between pairs of fiducial points.
To assess the influence of inter-subject variability, breathing pattern, and posture, the nonparametric Kruskal-Wallis test was applied to each feature and fiducial point.The χ 2 statistic and its corresponding significance were assessed to determine which fiducial points were most prone to influence by each of the three controlled factors.
For the ability of the different fiducial points to provide accurate measurements of the RR time series, the differences between the IBI time series defined as in Eq (6) and the RR time series defined as RR(n) = R(n+1) -R(n) were analyzed for each fiducial point.Agreement between both time series was quantified from the standard deviation of the difference between time series that equals the differentiation of the PAT time series (PAT(n+1)-PAT(n)), as shown in Eq (6).The standard deviation of the differentiated PAT time series (σ RR-IBI ) was analyzed in the same way as the previous features were.
The p<0.001 was considered to indicate statistical significance, while p<0.05 was considered to indicate statistical significance.

Results
Table 1 shows the median, 90%, and 10% percentiles of α opt and the percentage of times that tag opt is positive (indicating that the fiducial point corresponds to a maximum) according to the optimization criterion.Paired sign tests revealed that, compared with those of either the maxagr or minSDHPF, the α opt and tag opt of the minSDPAT were significantly different.Nevertheless, no significant differences were detected for α opt or tag opt when comparing maxagr and minSDHPF.According to unpaired sign tests, the values of α opt showed a median very statistically significant difference of zero for maxagr, but the difference was not statistically significant for α opt when using the minSDPAT or minSDHPF optimizations.On the other hand, the values of tag opt show a highly statistically significant difference of zero for maxagr and minSDHPF but not for minSDPAT.A nonparametric Fisher test showed that there were very statistically significant nonrandom associations between the sign of α opt and tag opt (77.0%chance of having the same sign for maxagr, 82.6% for minSDPAT and 74.7% for minSDHPF).
Table 2 shows the χ 2 statistics as the result of nonparametric Kruskal-Wallis tests on the values of α opt and tag opt assessing the effect of the three controlled factors (subject, breathing rate, and posture) disclosed by the optimization criterion compared by the three controlled factors.The results showed that the values of α opt and tag opt are strongly dependent on the individual and the posture measured but are not associated with the breathing pattern.The median α opt values for maxagr were -0.76 while sitting (N = 120) and -0.07 while standing (N = 58); for minSDPAT, the values were -0.38 and 1.73, respectively; and for minSDHPF, the values were -0.60 and 0.91, respectively.Hence, α opt tends to be lower when sitting than when standing.The chance that tag opt is equal to 1 for maxagr is 28.3% while sitting and 46.6% while standing; for minSDPAT, it is 45.8% and 65.5%, respectively; and for minSDHPF, it is 25.0% and 51.7%, respectively.Hence, the fiducial point is more likely to be at its maximum while standing than while sitting.
The median percentage of outliers in the PAT time series was zero for all the fiducial points except for P peak (0.31%), indicating that at least 50% of the recordings were free of any outliers for most fiducial points.The mean percentage of outliers was 0.67% for P rough , 1.32% for P peak , 1.21% for P foot , 0.35% for P pfd , 0.73% for P psd , 0.43% for P tanh , 0.33% for P tan , 0.25% for P maxagr , 0.23% for P minSDPAT and 0.25% for P minSDHFPAT .The 95% percentiles for the percentages of outliers were 3.44% for P rough , 6.62% for P peak , 4.40% for P foot , 2.69% for P pfd , 1.87% for P psd , 1.40% for P tanh , 1.87% for P tan , 1.61% for P maxagr , 1.57% for P minSDPAT and 1.61% for P minSDHFPAT .3 and 4, the mean PAT changes depending on the choice of fiducial point.Moreover, standard deviation and breathing activity could be influenced by fiducial point choice.For example, the mean PAT will reach a minimum when using the foot of the pulse wave and will reach a maximum when using the peak of the pulse wave, while the remaining fiducial points will have a mean PAT within.Fig 4 shows that the PAT with the lowest standard deviation using the optimal fractional differintegration fiducial point with the minSDPAT optimization criterion also had the most apparent contribution to breathing activity when compared with the other two criteria (the subject was breathing at 15 bpm in this particular example).
Table 3 shows the median, 10th percentile, and 90th percentile of the mean and standard deviation of the PAT time series as well as for the ResprPAT index and the standard deviation of the difference between the RR time series and the IBI time series.The results are disclosed by the fiducial point.Table 4 shows the z values of the paired sign tests for the features for each pair of fiducial points.

Discussion
This work has aimed to provide a new versatile tool for determining the arrival of pulse waves.We propose three different optimization criteria to obtain fiducial points, define the PAT time series (and, accordingly, the IBI time series), and compare the results with other fiducial points that can be found in the literature.Of course, other optimization criteria can be provided to optimize some kind of feature to suit the researcher's needs.For example, if two pulse waves (one downstream of the other) are used instead of using the ECG signal and the pulse wave at a distal site, a feasible optimization criterion will be to minimize the standard deviation of the differentiation of the successive differences in pulse arrivals.Using this procedure (which implies optimizing PA a opt tag opt for the proximal and distal pulse waves), a smooth PTT time series will be obtained that suits the hypothesis that changes in pulse wave velocity cannot have highfrequency contents because they are mediated by the sympathetic nervous system.In the remainder of this section, we will discuss only the results and their implications for the three employed optimization criteria.
The results in Table 1 and associated tests show that the values of α opt and tag opt are dependent on the choice of the optimization criteria, although maxagr and minSDHFPAT behave quite similarly with a high probability.The pulse wave is fractionally integrated (negative value of α opt ), and the fiducial point then corresponds to the minimum during the systolic rise.Both criteria provide similar results because maxagr minimizes the standard deviation of the differentiated PAT time series, as shown in Eq (6), while minSDHFPAT minimizes the standard deviation of the fast (high-frequency band) changes in the resampled PAT time series.Both criteria aim to reduce similar factors.In contrast, the minSDPAT aims to minimize the standard deviation of any change in the PAT time series regardless of the spectral content of these changes.This criterion has been proposed for comparison with the performance of traditional fiducial points, especially when P tanh claims to provide a minimum standard deviation of the PAT time series when compared with other traditional fiducial points.For minSDPAT, the optimal fiducial point can be obtained either by fractionally integrating or by fractionally differentiating the pulse wave.When using fractional integration, the chances that the fiducial point is the minimum are even greater than those for the other two optimization criteria.Moreover, when fractionally differentiating, the chances that the fiducial point corresponds to a maximum are also very high.
Table 2 shows that the values of α opt and tag opt depend mostly on the subject of interest.The posture has a mild influence on these values, while breathing has no statistical effect on them.The inter-subject variability is also an issue for the features of the PAT time series for all the analyzed fiducial points, as shown in Table 5.These changes among subjects can be explained by differences in the recorded pulse waveforms.Nevertheless, a change in waveform from one subject to another can be caused by changes in hemodynamics or changes in sensor attachment.Further research using a multi-sensor approach or deliberately changing the hemodynamics of the subject will be performed to better understand the role of the inter-subject variability in the values of α opt, tag opt and different PAT features.The first column in Table 3 shows that the mean value of PAT depends, of course, on the fiducial point choice.According to our database, the median value of this feature reaches a minimum for P foot , which is normal because it defines the start of the systolic increase and has a maximum for P peak , which marks the end of the systolic increase.In between and for increasing values of the mean of the PAT, we have P psd , P tan , P minSDPAT , P maxagr , P pfd , P tanh , P minSDHF- PAT, and P rough .Therefore, the large change in PAT associated with the fiducial point limits comparisons of results among studies with different fiducial point selections.On the other hand, the 10% to 90% percentile interval for PAT is greater for the fractionally differintegrated derived fiducial points than for the traditional fiducial points.The only two other fiducial points showing an interval greater than 100 ms are P rough and P peak .Nevertheless, the causes of this high variability among these fiducial points obey different mechanisms.The final portion of the systolic increase slowly changes, where P rough and P peak lie.Any subtle change in morphology among recordings will explain the high variability in PAT when using P peak or P rough .On the other hand, the high variability in PAT when using P maxagr , P minSDPAT, or P minSDHFPAT obeys the need to lie in some place between the start and the end of the systolic rise to suit the optimization criteria.Hence, the location is suitable for each recording.For some recordings and optimization criteria, the fractional differintegrated derived fiducial points can even reach the location of the foot or the peak of the systolic rise.The first block in Table 4 shows that, of course, P peak provides a PAT with a median value very significantly greater than the rest of the fiducial points, while P foot provides a value very significantly lower than the rest of the fiducial points.The only fiducial points where there were no significant differences when comparing the median PAT using P pfd or P maxagr , when using P tan or P minSDPAT, and when using P minSDHFPAT or either P pfd , P tanh or P maxagr .These test results suggest that the most likely locations of P maxagr and P minSDHFPAT are near P pfd and P tanh respectively, close to the midpoint of the systolic rise, while P minSDPAT is most likely located in the initial portion of the systolic rise, close to P tan .Nevertheless, the actual location depends on the recording.To finish the (P pfd , P tanh, P maxagr, and P minSDHFPAT ) and very significantly affects the feature of fiducial points located near the end of the systolic rise (P rough and P peak ).Morphological changes due to hydrostatic pressure changes and the physiological response to compensate for these changes may explain this effect.Nevertheless, further experimentation is needed to confirm whether these changes are associated with posture.Table 3 shows the results for the median and the 10th to 90th percentiles of σ PAT for the different fiducial points, while the second block of Table 4 shows the results for testing for significant differences in the median values of the paired samples.As expected by its optimization criterion, the fiducial point that provides the PAT time series with the lowest standard deviation is P minSDPAT , which shows very significant differences from the rest of the fiducial points.To increase the median value of σ PAT come P tan , P maxagr, P minSDHFPAT , P foot , P psd , P tanh , P pfd, P rough and, as expected, P peak because of its sensitivity to noise.The performance of P tanh is quite deceiving because Solà J et al. [26] claimed that it is a fiducial point that provides a low standard deviation PAT.These differences may be attributable to the differences in the recording sampling frequencies and experimental setup used to test the performance.Once again, σ PAT is mostly influenced by the measured subject so it can be employed as a physiological index characterizing them.In contrast, posture seems not to affect value, and the effect of breathing can be significant depending on the employed fiducial point.σ PAT , defined by fractional differintegrated derived fiducial points, seems not to be affected by breathing or P peak .Although for P peak this result can be attributed to the dominant noise effect in σ PAT , for P maxagr , P minSDPAT , and P minSDHFPAT , we hypothesize that this insensitivity to breathing stems from the ability of the algorithm to find fiducial points less affected by morphological changes associated with breathing.Nevertheless, further analysis of optimally fractional differintegrated pulse waves is needed to test this hypothesis.However, the effect of breathing on the rest of the fiducial points was significant but not as significant as the effect on the inter-subject variability.
For ResprPAT, the results follow a similar pattern as for σ PAT .Table 3 shows that breathing is a high-frequency process, and the lowest ResprPAT corresponds to P minSDHFPAT .Next, they included P maxagr , P minSDPAT , P tan , P psd , P tanh , P peak , P pfd , P foot and P rough .The most sensitive traditional fiducial points for breathing are most likely located in the middle to last section of the systolic rise.The PAT, when using the P foot test, also has a greater modulation associated with breathing.The third section of Table 4 shows that ResprPAT is very significantly lower for P minSDHFPAT than for the other studied fiducial points, as predicted by the employed optimization criterion.Inter-subject variability is the factor that most affects the ResprPAT, while posture seems not to influence its value.Depending on the fiducial point, breathing can have a very significant influence or a mildly significant influence (this is the case for P minSDPAT and P psd ).These influences suggest that the variability in the PAT associated with breathing is dependent on the respiratory rate.
For the agreement between the RR and the IBI time series, the quantifier σ RR-IBI is, as foreseeable, the minimum when using P maxagr as shown in Table 3.The fourth block in Table 4 also shows that this fiducial point shows very significant differences, with the rest of the fiducial points being σ RR-IBI consistently lower than when using other alternatives.After P maxagr , the best agreement is obtained for P minSDHFPAT , followed by P minSDPAT and P tan .Note that for optimization purposes, the fractional differintegration derived fiducial points require knowledge of the R wave location to define the fiducial point location although the remaining fiducial points do not need to define the IBI time series.Hence, P tan is, using our database, the best fiducial point for defining a time series with likely good agreement with the RR time series without the need to know the R peak locations.After P tan , sorted by increasing σ RR-IBI , come to P tanh , P foot , P psd , P pfd , P rough , and, finally and not surprisingly, P peak .Table 5 shows that intersubject variability is the most influencing factor on agreement, while breathing and posture have small effects.As is the case for σ PAT ,σ RR-IBI can be used as a physiological marker for the subjects being measured.
In HRV measurements, two time domain indices are commonly used in the literature [38]: the SDNN, which corresponds to the standard derivation of the RR time series, and the rmsDD, which corresponds to the standard deviation of the first differentiation of the RR time series.Note that σ PAT is the same index as SDNN but is applied to the PAT time series, while σ RR-IBI is similar to rmsDD for the PAT time series in accordance with Eq (6).Nevertheless, although the different indices in HRV analysis are slightly affected by the fiducial point employed to define the R wave (because of its impulsive nature), the analysis of PAT variability is strongly influenced by the fiducial point choice, as the measured values of PAT,σ PAT ResprPAT, and σ RR-IBI can significantly change among fiducial points.
As shown in the results, all the analyzed features and optimization procedures depend on the subject under measurement (and probably how the sensors are attached).Hence, our results can change by expanding the database of interest.Future work will include other physiological databases that include simultaneous measurements of ECG and PPG signals to test how the results change for different experimental setups (e.g., PPG location, subject posture, sampling frequency, and movement restrictions).Although, for the setup of our measurements, the number of outliers was kept low, the results change slightly by modifying the outlier detection criterion.It is foreseeable that the results would change for other databases with a large number of outliers.Analysis of new databases could also serve as a way to test whether the proposed new set of fiducial points is robust to noisy events such as movement or poor sensor attachment.
Finally, the PAT has been proposed as a surrogate measure of PTT that is strongly correlated with blood pressure.As indicated in the introductory section, some studies acknowledge the utility of PAT for noninvasively estimating blood pressure, but others indicate that these blood pressure estimations are not accurate enough.These differences in accuracy may be caused by the choice of fiducial point.These studies focused on estimations of the average value of blood pressure (both systolic and diastolic) from the mean values of either the PAT or PTT.We hope that beat-to-beat analysis of PAT could provide a better way to noninvasively track the dynamics of blood pressure if the right fiducial point is selected.Because the regulation of pulse wave velocity and blood pressure is dominated by low-frequency processes, we hypothesize that optimization criteria such as minSDHFPAT or maxagr, which try to minimize the high-frequency of PAT, may work better in the quest for beat-to-beat blood pressure estimation.Hence, further experimentation with direct measurements of beat-to-beat blood pressure or beat-to-beat pulse wave velocity could refute or compromise the utility of this new set of fiducial points.

Conclusions
This paper has proposed a new set of fiducial points based on the fractional differintegration of pulse waves to improve the detection of pulse arrivals.As the set of fiducial points is infinite, several optimization criteria can be proposed to improve detection based on different needs.This paper has proposed three optimization criteria to maximize the agreement of the IBI time series with the RR time series, to minimize the standard deviation of the PAT time series, or to minimize the standard deviation of the high-frequency contents of the PAT time series.To assess the performance of the proposed method, this study investigated seven traditional fiducial points commonly used in PAT and IBI and compared their performance against a newly proposed set of fiducial points.The chosen characteristics for evaluating the fiducial points included accurate pulse arrival detection, consistency with the RR time series, and robustness to variability.The best-performing fiducial points showed accurate pulse arrival detection, alignment with the RR time series, and robustness to variations in pulse wave morphology.In contrast, the worst-performing fiducial points exhibit poor adaptability to physiological changes, including posture changes, inaccurate pulse arrival detection, and vulnerability to external factors such as breathing patterns.The results showed that our proposal can provide an IBI time series with better agreement with the RR time series rather than other traditional fiducial points (median agreement of 5.72 ms in contrast with 6.62 ms for P tanh , 7.29 ms for P foot or even 9.93 ms for P peak ); a PAT series with a lower standard deviation than other traditional methods (median standard deviation of 7.21 ms versus 8.67 ms for P tanh , 8.22 ms for P foot or 15.4 ms for P peak ); and a PAT series with lower respiratory modulation than other fiducial points (median respiratory contribution of 3.44 ms versus 4.50 ms for P tanh , 4.72 ms for P foot or 4.56 ms for P peak ).
According to this result, the proposed fiducial points showed a notable reduction in the standard deviation of the PAT compared to traditional points, suggesting its potential for a more accurate assessment of beat-to-beat PAT variability and IBI time series.
Further research may define better optimization criteria to improve the measurement of beat-to-beat PAT variability as a surrogate measurement of the dynamics of the pulse wave velocity (PWV).

Fig 1 .
Fig 1. Example of fractional differintegrated versions of a pulse wave.The blue line shows a pulse wave encompassing around a cardiac cycle, while the red line corresponds to the systolic increase in the pulse wave.The dashed lines correspond to the application of fractional differintegrations to the signal using an arbitrary α (the values are shown in the legend).All signals were normalized between 0 (minimum) and 1 (maximum).https://doi.org/10.1371/journal.pone.0298354.g001

Fig 3
shows an example of a measured PAT time series defined for different fiducial points, while Fig 4 compares the PAT time series using the optimal fiducial points obtained by fractional differintegration for the same recording of Fig 3 but using the three different optimization criteria.As shown in Figs

Fig 3 .
Fig 3. PAT time series obtained from different fiducial points defined during the systolic rise of the pulse wave.For the definition of the optimal fractional differintegration fiducial point, the maxagr optimization criterion was employed.https://doi.org/10.1371/journal.pone.0298354.g003

Fig 4 .
Fig 4. PAT time series for the same example shown in Fig 3.It is obtained from optimal fractional differintegration derived fiducial points using the three proposed optimization criteria.https://doi.org/10.1371/journal.pone.0298354.g004

Table 2 . Results of the Kruskal-Wallis nonparametric test comparing the medians of variables for several groups.
The table shows the χ 2 statistic and its significance (n not significant, † p<0.05, ‡ p<0.001).The factors of subject, breathing and posture were tested separately. https://doi.org/10.1371/journal.pone.0298354.t002

Table 3 . Descriptive statistics for PAT time series characterization and agreement of IBI with RR time series disclosed by fiducial point.
The values are the medians and the [10%-90%] percentiles of the indices.

Table 5 . Results of the Kruskal-Wallis nonparametric test comparing the medians of several variables.
The table shows the χ 2 statistic and its significance ( † p<0.05, ‡ p<0.001).The factors subject (S), breathing (B) and posture (P) were tested separately.The tests are applied for the different fiducial points.